Method for determining future position boundary for a moving object from location estimates

ABSTRACT

A method for estimating a boundary for a future location of a moving object includes: receiving location information about two previous locations of the moving object represented by a first and a second elliptical error probabilities (EEPs); representing each of the EEPs as a conic section via an implicit quadratic equation; defining four tangent lines from the implicit quadratic equation, each tangent line being tangent to both of the EEPs; determining two transverse tangent lines from the four tangent lines; forming a cone in a direction from the first EEP to the second EEP from the two transverse tangent lines; and estimating the boundary of the future location of the moving object as a first side of the cone formed by a first transverse tangent line and a second side of the cone formed by a second transverse tangent line, of the two transverse tangent lines.

FIELD OF THE INVENTION

The present invention relates generally to location estimation; and moreparticularly to a method for determining future position boundary for amoving object from location estimates.

BACKGROUND

Typically, position/location estimates include some errors. For example,RF trilateration methods use estimated ranges from multiple receivers toestimate the location of an object. RF triangulation uses the angles atwhich the RF signals arrive at multiple receivers to estimate thelocation of the object. However, many obstructions, such as barriers,clouds, landscape objects, and the like can distort the estimated rangeand angle readings leading to varied qualities of location estimate.Estimation-based locating is often measured in accuracy for a givenconfidence level. In other words, how frequently an observed intervalcontains the desired parameter is determined by the confidence level(confidence coefficient). More specifically, if confidence intervals areconstructed across many separate data analyses of repeated (and possiblydifferent) experiments, the proportion of such intervals that containthe true value of the parameter will match the confidence level.

A confidence region is a multi-dimensional generalization of aconfidence interval, that is, a set of points in an n-dimensional space,which is often represented as an ellipsoid around a point which is anestimated solution to a problem, for example, a set of locationestimates. In a two-dimensional space, confidence region is representedas an ellipse. The confidence region is calculated in such a way that ifa set of measurements were repeated many times and a confidence regioncalculated in the same way on each set of measurements, then a certainpercentage of the time, on average the confidence region would includethe point representing the “true” values of the set of variables beingestimated, for example, a set of location estimates. Such an ellipticalconfidence region is conventionally referred to as an elliptical errorprobability (EEP).

Current position estimation approaches do not predict a future positionof a moving object. Instead, they typically only provide a pointestimate of the location along with parameters of an EEP at a particularlevel of confidence (e.g. 95%).

Moreover, in many situations, it is impractical to directly observewhere a moving object is and to where it may be traveling.

Accordingly, there is a need for a method for determining the futureposition boundary of a moving object, which utilizes the alreadydetermined two previous location estimates. Each of the two previouslocation estimates being represented by an elliptical error probability(EEP).

SUMMARY

In some embodiments, the computer implemented method of the presentinvention capitalizes upon geo-located signal (e.g., RF, sonar, radar,and the like) emissions generated by a moving object, and certainassumptions, to establish the spatial bounds within which the movingobject may be traveling. Geolocation of signal emissions, usuallyderived from signals sources, is a well-established capability. Itprovides, with a level of confidence, a knowledge of where the emitterof a signal is located. The present invention provides a new capabilityto establish the spatial bounds, with a level of confidence, for where amoving signal emitter may be going, based upon its geo-located signalemissions that are received over time. This provides useful informationfor target tracking and pattern-of-life analysis of moving signalemitters (objects) for a variety of applications, such as Activity BasedIntelligence and data fusion applications.

In some embodiments, the present invention is a computer implementedmethod for estimating a boundary for a future location of a movingobject. The method includes: receiving location information about twoprevious locations of the moving object represented by a first and asecond elliptical error probabilities (EEPs); representing each of thefirst and the second EEPs as conic sections via implicit quadraticequations; defining four tangent lines from the implicit quadraticequations, each tangent line being tangent to both of the first and thesecond EEPs; determining two transverse tangent lines from the fourtangent lines; forming a cone in a direction from the first EEP to thesecond EEP from the two transverse tangent lines; and estimating theboundary of the future location of the moving object as a first side ofthe cone formed by a first transverse tangent line and a second side ofthe cone formed by a second transverse tangent line, of the twotransverse tangent lines.

In some embodiments, the present invention is a tangible computerstorage medium having stored therein computer instructions. The computerinstructions when executed by one or more computers perform: receivinglocation information about two previous locations of the moving objectrepresented by a first and a second elliptical error probabilities(EEPs); representing each of the first and the second EEPs as conicsections via implicit quadratic equations; defining four tangent linesfrom the implicit quadratic equations, each tangent line being tangentto both of the first and the second EEPs; determining two transversetangent lines from the four tangent lines; forming a cone in a directionfrom the first EEP to the second EEP from the two transverse tangentlines; and estimating the boundary of the future location of the movingobject as a first side of the cone formed by a first transverse tangentline and a second side of the cone formed by a second transverse tangentline, of the two transverse tangent lines.

The cone then may be superimposed over a map to visualize the bounds forthe probable future location of the moving object.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows spatial bounds of the straight line travel at differentconfidence levels, according to some embodiments of the presentinvention.

FIG. 2 depicts one possible trajectory of a moving object from a firstellipse to a second ellipse, according to some embodiments of thepresent invention.

FIG. 3 shows two error ellipses and an interconnecting path within aCartesian coordinate system, according to some embodiments of thepresent invention.

FIG. 4 depicts boundaries of probable straight-line path of a movingobject travelling through two ellipses, according to some embodiments ofthe present invention.

FIG. 5 shows four tangential lines between two ellipses, according tosome embodiments of the present invention.

FIG. 6 graphically illustrates that the traverse tangent lines betweenthe two ellipses provide the maximum angle formed by a vector passingfrom the first ellipse to the second ellipse, according to someembodiments of the present invention.

FIG. 7 is an exemplary process flow, executed by one or more computers,according to some embodiments of the present invention.

DETAIL DESCRIPTION

Although the present invention is described in the context of a computerexecutable code, one skilled in the art would recognize that theinvention may be implemented as software for a general purpose computer,firmware for the special purpose computer, or a combination thereof. Insome embodiments, the present invention determines the spatial bounds ata given confidence level, in which a moving object (such as ships,aircrafts, land vehicle, and the like) might exist at future times froma pair of given location estimates in the form of confidence ellipses,under the assumption of constant speed and straight line motion. Suchbounds are expected to resemble a “hurricane track”, as shown in FIG. 1.In FIG. 1, the pairs of lines denote position bounds at varying levelsof confidence (1−α), with the innermost pair being actually a singleline connecting the center of the two ellipses.

FIG. 7 is an exemplary process flow for estimating a boundary for afuture location of a moving object, executed by one or more computers,according to some embodiments of the present invention. A shown in block702, location information about two previous locations (or a previousand a current location) of the moving object is received. Thisinformation is represented by a first elliptical error probability (EEP)and a second EEP. Each of the EEPs may be represented by a centroid, asemi-major axis, a semi-minor axis, and a rotation angle. Each EEPincludes a mean location in the form of a two dimensional vector inlatitude/longitude and a covariance matrix. In block 704, each of thefirst and the second EEPs is represented in a quadratic form. This isaccomplished by converting the parameters of the EEP to parameters ofthe quadratic form using well-known conversion formulas. Themathematical derivation of how an ellipse's characteristics areconverted to the parameters for the corresponding conic section, i.e.the parameters (a, b, c, d, e, & f) of the quadratic equation isdescribed in Appendix A, the entire content of which is hereby expresslyincorporated by reference.

In block 706, four tangent lines that are tangent to both of the EEPsare defined, from the implicit quadratic equations. Two of the fourtangents, known as direct tangential lines, do not provide the desiredmaximally divergent boundary lines. The other two tangent lines known astransverse tangential lines are of interest. The two traverse tangentlines between the two EEPs provide the maximum angle formed by a vectorpassing from the first EEP to the second EEP. Each tangent line has twopoints, for example, (x₁, y₁) and (x₂, y₂) that lie on the first andsecond EEPs, respectively. These four unknowns x₁,y₁,x₂, y₂, can besolved by four equations that specify these four unknowns. In block 708,the two transverse tangent lines are determined from the four tangentlines.

In some embodiments, the direct and transverse tangents are identifiedby using the knowledge that each pair of tangent points constitutes theend points of a line segment. Thus, for any two line segments torepresent transverse tangential lines, they must cross each other. Linecrossing can be detected when each tangent point of a tangent line fallson an opposite sides of the line it supposedly crosses. This can bedetermined through the two-dimensional cross product between the line,as defined by and (x₁,y₁) and (x₂,y₂) the point (x,y):

(x ₂ −y ₁)(y−y ₁)−(y ₂ −y ₁)(x−x ₁)

If the result is positive, the point is on one side of the line. If theresult is negative, the point is on the other side of the line. If theresult is zero, the point is on the line. This test is performed foreach tangent line using another tangent line to determine whether theycross each other. Those tangent lines that successfully pass the testare placed in a list. Nominally, there should be two transverse tangentlines in the list.

A cone is then formed in a direction from the first EEP to the secondEEP, from the two transverse tangent lines, in block 710. This is doneby constructing a cone whose vertex corresponds to the point ofintersection of the tangents and whose sides are the tangent linesegments contacting the second EEP. In block 712, the boundary of thefuture location of the moving object is estimated, for example, bysuperimposing the cone formed in block 710 over a map. This allows ahuman operator to visualize the potential future location of the movingobject.

FIG. 2 shows one possible trajectory of a moving object from a firstellipse to a second ellipse, according to some embodiments of thepresent invention. Under the well-known Bonferroni joint estimationapproach, for a confidence level of 1−α, a confidence level of 1−α/2 isrequired for each location estimate. Consider the (1−α/2)-levelconfidence ellipse E1 for a first location 1 and the (1−α/2)-levelconfidence ellipse E2 for a second location 2, that the object moves to,after the location 1. By assumption, the moving object's futuretrajectory can result from the combination of any point u in E1 and anypoint v in E2, shown as a dashed line in FIG. 2.

FIG. 3 shows two error ellipses and an interconnecting path within aCartesian coordinate system, according to some embodiments of thepresent invention. Consider a point (x₁,y₁) in E1 and a point (x₂,y₂) inE2 and the point y where the corresponding heading intersects the linex. Now, for a particular value of x, the solution corresponding to themaximum and minimum values of y comprises of the two y values thatcorrespond to the limits of headings intersecting the line x. Denotingthese extreme y values y and for the lower and upper limit,respectively, the problem is thus formulated as follows:

$\underset{\_}{y} = {\min\limits_{\underset{{({x_{2},y_{2}})} \in {E\; 2}}{{({x_{1},y_{1}})} \in {E\; 1}}}y}$$\overset{\_}{y} = {\min\limits_{\underset{{({x_{2},y_{2}})} \in {E\; 2}}{{({x_{1},y_{1}})} \in {E\; 1}}}y}$

The general, implicit form of an ellipse is given by ax²bxy+cy²dx+ey+f=0 subject to the constraint b²−4ac<0.

Note that this representation can be obtained by the more familiarmean-covariance representation at confidence level κ having parameters:

$\mu = \begin{bmatrix}x_{0} \\y_{0}\end{bmatrix}$ $\rho = \begin{bmatrix}\rho_{11} & \rho_{12} \\\rho_{12} & \rho_{22}\end{bmatrix}$

via the following

a=ρ₂₂

c=ρ₁₁

b=2ρ₁₂

d=2(ρ₁₂y₀−ρ₂₂x₀)

e=2(ρ₁₂x₀−ρ₁₁y₀)

f=κ(ρ₁₂ ²−ρ₁₁ρ₂₂)+ρ₂₂x₀ ²+ρ₁₁y₀ ²−2ρ₁₂x₀y₀

where μ a represents the ellipse center and p represents the ellipsecovariance.

Next, the invention, finds traverse tangents from the maximal angles.FIG. 4 depicts boundaries of probable straight-line path of a movingobject travelling through two ellipses, according to some embodiments ofthe present invention. A positional error ellipse indicates theiso-density of a bivariate normal probability density function for thelocation of the object. Let a moving object travel on a vector startingsomewhere within error ellipse E1 and passing through error ellipse E2,as illustrated in FIG. 4. Let E1 be an ellipse centered at the origin ofa coordinate system and rotated by some arbitrary amount. Let E2 beanother ellipse offset from the origin by some arbitrary amount, withits center on the x axis and also rotated by some arbitrary amount. Theprobable vector of the moving object will lie between two extrema thatare bounded by all possible vectors that start from E1 and passingthrough E2. It is desired to find the maximum values for angle PL₁ (withvertex at a) and angle PL₂ (with vertex at b) as these represent themaximum headings under which an object at those points could be moving.

It would appear that the intersect points of the confidence vectors onthe surfaces of the error ellipses must be tangential in order toprovide the maximum extent of deviation of the confidence vectors. Aproof of this is provided below.

FIG. 5 shows four tangential lines between two ellipses, according tosome embodiments of the present invention. In order to derive these twoboundaries, the general case of tangent lines between two ellipses areconsidered, as illustrated in FIG. 5. The lines L₁ and L₂, which are ofinterest, are known as transverse tangential lines. The lines L₃ and L₄are known as direct tangential lines, which do not provide the desiredmaximally divergent border lines. Each tangent line has two points, forexample, (x₁, y₁) and (x₂, y₂) for line L₁, that lie on the ellipses E1and E2, respectively. The four unknowns, named hereinafter x₁,y₁,x₂,y₂,can be solved if four equations can be formulated that specify thesefour unknowns.

FIG. 6 graphically illustrates that a traverse tangent line between thetwo ellipses provide the maximum angle formed by a vector passing fromthe first ellipse to the second ellipse, according to some embodimentsof the present invention. As shown, the centers of two non-intersecting,arbitrarily rotated ellipses E1 and E2 are on a line P. It is desired toshow that the maximum divergence angles from P are defined by thetransverse tangential lines joining E1 and E2. Note that for clarity,only one of the two transverse tangential lines is shown.

The two-point form of a line can be expressed by the equation shownbelow:

−(y ₂ −y ₁)x+(x ₂ −x ₁)y+x ₁ y ₂ −x ₂ y ₁=0.

A point (x_(n),y_(n)) that is not on the line can be described throughinequalities. If the point is to the left of the line, then theinequality is

−(y ₂ −y ₁)x _(n)+(x ₂ −x ₁)y _(n) +x ₁ y ₂ −x ₂ y ₁=0.

If the point is to the right of the line, then the inequality is

−(y ₂ −y ₁)x _(n)+(x ₂ −x ₁)y _(n)+x₁ y ₂ −x ₂ y ₁=0.

A tangent line that intersects point (x₁,y₁) on ellipse E1 in FIG. 6,forms a bound that all other points on E1 must lie to the left of thatline. Likewise, a tangent line that intersects point (x₂,y₂) on ellipseE2 (FIG. 6) forms a bound that all other points on E2 must line to theright of that line. If points (x₁,y₁) and (x₂,y₂) share the same tangentline, then any line L′₁ connecting any two points (x′₁,y′₁) and(x′₂,y′₂), in E1 and E2 respectively, must intersect line L₁. For L′₁ tointersect L₁, the slope of L′₁ must be different than the slope of L₁.

Given the restriction that (x′₁,y′₁) must be in E1 and (x′₂,y′₂) must bein E2, the slope of L′₁ must be less than the slope of L₁. Therefore,any angle formed by a line PL′₁ must be less than the angle formed byPL₁, which is then the maximum divergence angle from P. For line L₂ inFIGS. 4 and 5, this proof also applies, with the slope of an imaginedvector L′₂ being greater (less negative) than the slope of L₂.

The invention then defines a point on a conic section in order toestablish equations that can be solved to find the desired tangents. Toobtain the elliptical quadratic form corresponding to an EEP, a point ona conic section needs to be defined. The general, implicit definitionfor a conic curve is

ax ² +bxy+cy ² +dx+ey+f=0

For the condition b²−4ac<0, the conic curve forms an ellipse. The points(x₁,y₁) and (x₂,y₂) on ellipses E1 and E2, respectively, can then bespecified through separate equations for these ellipses.

a ₁ x ₁ ² +b ₁ x ₁ y ₁ +c ₁ y ₁ ² +d ₁ x ₁ +e ₁ y ₁ +f ₁=0

a ₂ x ₂ ² +b ₂ x ₂ y ₂ +c ₂ y ₂ ² +d ₂ x ₂ +e ₂ y ₂ +f ₂=0

Since the desired mover boundaries are within the tangent lines, thetangent lines of the two EEP need to be determined. In order to find thetangent line on any point in the ellipse, implicit differentiation mustbe performed

${{2{ax}} + {b\left\lbrack {y + {x\frac{y}{x}}} \right\rbrack} + {2{cy}\frac{y}{x}} + d + {e\frac{y}{x}}} = 0.$

Isolating the differential terms on the left side of the e uation yields

${\frac{y}{x}\left\lbrack {{bx} + {2{cy}} + e} \right\rbrack} = {{{- 2}{ax}} - {by} - d}$

The equation then becomes

$\frac{y}{x} = \frac{{{- 2}{ax}} - {by} - d}{{bx} + {2{cy}} + e}$

The tangential line formed by (x₁,y₁) and (x₂,y₂) has the slope

$\frac{y_{2} - y_{1}}{x_{2} - x_{1}}.$

Therefore, two equations can be stated that relate the slope of thetangent lines on E1 and E2 to the slope of the shared line.

$\frac{{{- 2}a_{1}x_{1}} - {b_{1}y_{1}} - d_{1}}{{b_{1}x_{1}} + {2c_{1}y_{1}} + e_{1}} = \frac{y_{2} - y_{1}}{x_{2} - x_{1\;}}$$\frac{{{- 2}a_{2}x_{2}} - {b_{2}y_{2}} - d_{2}}{{b_{2}x_{2}} + {2c_{2}y_{2}} + e_{2}} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

These equations can be reordered, bringing the denominators out fromunder their respective numerators.

(−2a ₁ x ₁ −b ₁ y ₁ −d ₁)(x ₂ −x ₁)=(b ₁ x ₁+2c ₁ y ₁ +e ₁)(y ₂ −y ₁)

(−2a ₂ x ₂ −b ₂ y ₂ −d ₂)(x ₂ −x ₁)=(b ₂ x ₂+2c ₂ y ₂ +e ₂)(y ₂ −y ₁)

Multiplying out the terms yields:

2a ₁ x ₁ ² +b ₁ x ₁ y ₁ +d ₁ x ₁−2a ₁ x ₁ x ₂ −b ₁ x ₂ y ₁ −d ₁ x ₂ =b ₁x ₁ y ₂+2c ₁ y ₁ y ₂ +e ₁ y ₂ −b ₁ x ₁ y ₁−2c ₁ y ₁ ² −e ₁ y ₁

2a ₂ x ₂ +b ₂ x ₁ y ₂ +d ₂ x ₁−2a ₂ x ₁ ² −b ₂ x ₂ y ₂ −d ₂ x ₂ =b ₂ x ₂y ₂+2c ₂ y ₂ ² +e ₂ y ₂ −b ₂ x ₂ y ₁−2c ₂ y ₁ y ₂ −e ₂ y ₁

Bringing all terms to the left-hand sides of their equations yields:

2a ₁ x ₁ ²+2b ₁ x ₁ y ₁ +d ₁ x ₁−2a ₁ x ₁ x ₂ −b ₁ x ₂ y ₁ −d ₁ x ₂ −b ₁x ₁ y ₂−2c ₁ y ₁ y ₂ −e ₁ y ₂+2c ₁ y ₁ ² +e ₁ y ₁=0

−2a ₂ x ₂ ²−2b ₂ x ₂ y ₂ −d ₂ x ₂+2a ₂ x ₁ x ₂ +b ₂ x ₁ y ₂ +d ₂ x ₁ +b₂ x ₂ y ₁+2c ₂ y ₁ y ₂ +e ₂ y ₁−2c ₂ y ₂ ² −e ₂ y ₂=0

The equations derived in the sections above can be brought together toprovide four equations with four unknowns.

a ₁ x ₁ ² +b ₁ x ₁ y ₁ +c ₁ y ₁ ² +d ₁ x ₁ +e ₁ y ₁ +f ₁=0  4

a ₂ x ₂ ² +b ₂ x ₂ y ₂ +c ₂ y ₂ ² +d ₂ x ₂ +e ₂ y ₂ +f ₂=0  5

2a ₁ x ₁ ²+2b ₁ x ₁ y ₁ +d ₁ x ₁−2a ₁ x ₁ x ₂ −b ₁ x ₂ y ₁ −d ₁ x ₂ −b ₁x ₁ y ₂−2c ₁ y ₁ y ₂ −e ₁ y ₂+2c ₁ y ₁ +e ₁ y ₁=0  6

−2a ₂ x ₂ ²−2b ₂ x ₂ y ₂ −d ₂ x ₂+2a ₂ x ₁ x ₂ +b ₂ x ₁ y ₂ +d ₂ x ₁ +b₂ x ₂ y ₁−2c ₂ y ₁ y ₂ −e ₂ y ₁+2c ₂ y ₂ ² −e ₂ y ₂=0  7

One approach to solving this system of multivariate polynomial equationsis to utilize Groebner bases. This approach has a property, which isgermane to the Transverse Tangents approach, called the “eliminationproperty”. The elimination property of Groebner bases indicates that thepolynomials of Groebner bases describe the indeterminates as a sequence.This allows for the solution of the system of polynomials “variable byvariable”. Consequently, the Groebner basis of a system of multivariatepolynomials can be used to find the solution set for the variables ofthat system. The Groebner basis of the system of multivariatepolynomials has the property that one of the equations is in terms of asingle variable. All possible values of that variable then can be easilyfound, thereby eliminating that equation. A second equation of theGroebner basis is stated in terms of the first variable and a secondvariable. Given that all possible values of the first variable can befound, all possible values of the second variable can be found also,thereby eliminating that equation as well. Each additional equation thatis part of the Groebner basis follows this same form. Consequently, allof the equations are ultimately eliminated and all possible values forall of the variables for the original system of multivariate polynomialscan be determined from the Groebner basis of that system.

It will be recognized by those skilled in the art that variousmodifications may be made to the illustrated and other embodiments ofthe invention described above, without departing from the broadinventive scope thereof. It will be understood therefore that theinvention is not limited to the particular embodiments or arrangementsdisclosed, but is rather intended to cover any changes, adaptations ormodifications which are within the scope of the invention as defined bythe appended claims.

1. A computer implemented method for estimating a boundary for a futurelocation of a moving object, the method comprising: receiving locationinformation about two previous locations of the moving objectrepresented by a first and a second elliptical error probabilities(EEPs); representing each of the first and the second EEPs as a conicsection via an implicit quadratic equation; defining four tangent linesfrom the implicit quadratic equation, each tangent line being tangent toboth of the first and the second EEPs; determining two transversetangent lines from the four tangent lines; forming a cone in a directionfrom the first EEP to the second EEP from the two transverse tangentlines; and estimating the boundary of the future location of the movingobject as a first side of the cone formed by a first transverse tangentline and a second side of the cone formed by a second transverse tangentline, of the two transverse tangent lines.
 2. The method of claim 1,further comprising superimposing the cone over a map to visualize thefuture location of the moving object.
 3. The method of claim 1, whereineach EEP includes a mean location in the form of a two dimensionalvector in latitude and longitude, and a covariance matrix.
 4. The methodof claim 1, wherein representing each of the EEPs as a conic section viaan implicit quadratic equation comprises converting parameters of theEEP to parameters of a quadratic form.
 5. The method of claim 1, whereindetermining two transverse tangent lines comprises representing each ofthe four tangent lines by a line segment; and determining line crossingof each tangent line with respect to the other three tangent lines bydetermining whether each tangent point of a tangent line falls onopposite sides of the line said each tangent line crosses.
 6. The methodof claim 1, wherein the two traverse tangent lines between the two EEPsprovide the maximum angle formed by a vector passing from the first EEPto the second EEP.
 7. The method of claim 1, wherein forming a conecomprises constructing a cone whose vertex corresponds to the point ofintersection of the two traverse tangent lines and whose sides are thetraverse tangent line segments contacting the second EEP.
 8. A tangiblecomputer storage medium having stored therein computer instructions, thecomputer instructions when executed by one or more computers perform:receiving location information about two previous locations of themoving object represented by a first and a second elliptical errorprobabilities (EEPs); representing each of the first and the second EEPsas a conic section via an implicit quadratic equation; defining fourtangent lines from the implicit quadratic equation, each tangent linebeing tangent to both of the first and the second EEPs; determining twotransverse tangent lines from the four tangent lines; forming a cone ina direction from the first EEP to the second EEP from the two transversetangent lines; and estimating the boundary of the future location of themoving object as a first side of the cone formed by a first transversetangent line and a second side of the cone formed by a second transversetangent line, of the two transverse tangent lines.
 9. The tangiblecomputer storage medium of claim 8, further comprising computerinstructions to perform superimposing the cone over a map to visualizethe future location of the moving object.
 10. The tangible computerstorage medium of claim 8, wherein each EEP includes a mean location inthe form of a two dimensional vector in latitude and longitude, and acovariance matrix.
 11. The tangible computer storage medium of claim 8,wherein representing each of the EEPs as a conic section via an implicitquadratic equation comprises converting parameters of the EEP toparameters of a quadratic form.
 12. The tangible computer storage mediumof claim 8, wherein determining two transverse tangent lines comprisesrepresenting each of the four tangent lines by a line segment; anddetermining line crossing of each tangent line with respect to the otherthree tangent lines by determining whether each tangent point of atangent line falls on opposite sides of the line said each tangent linecrosses.
 13. The tangible computer storage medium of claim 8, whereinthe two traverse tangent lines between the two EEPs provide the maximumangle formed by a vector passing from the first EEP to the second EEP.14. The tangible computer storage medium of claim 8, wherein forming acone comprises constructing a cone whose vertex corresponds to the pointof intersection of the two traverse tangent lines and whose sides arethe traverse tangent line segments contacting the second EEP.